450 Mr. J. J. Sylvester on a remarkable Modification 



integral or fractional, or mixed functions of a?, and whatever the 

 number of quotients (which, it may be observed, cannot be less 

 than, but may be made to any extent greater than the exponent 

 of the degree oif{a:)), will equally well furnish a Rhizoristic series 

 for fixing the position of the roots, provided only that the last 



divisor in the process of expanding ~- under the form of an 



improper continued fraction be a constant quantity or any func- 

 tion of a; incapable of changing its sign. 



Let us, however, for the present confine our attention to the 

 ordinary Sturmian form, where all the quotients are linear func- 

 tions of {x). Let these quotients be respectively 



fliar + ii; a^ + b^; a^-^b^; . . . anX + bn, 



In order to determine the total number of real and imaginary 

 roots of /(a:), we must count the loss of continuations of sign in 

 the Rhizoristic series in passing from a?= + ootoa7=-— go. When 

 X is infinitely great, it is clear that, whether positive or negative, 

 the parts 61, b^. . .bn may be neglected, and only the highest 

 powers of x need be attended to in writing down the signaletic 

 series corresponding to these two values of x. Accordingly for 

 a?= +x the signaletic series becomes -iiS:i» -jlU 



1, a^x^, a^a^^j ... a^ . «2 •••««• "2?" ; 



and consequently the number of pairs of imaginary roots oif{x) 

 in the number of changes of sign in the series 



1, «i, fll . «2, ...«]. flg^ . . . an, 



i. e, is the number of negative quantities in the series 



^v ^2^ % • • • «n- 

 Hence we have the curious and hitherto strangely overlooked 

 theorem, that in applying Sturm's process of successive division 

 to fx and fx, the number of negative coefficients of x in the 

 successive quotients gives the number of pairs of real roots of 

 fx; as a corollary, we learn the somewhat curious fact that never 

 more than half of these coefficients can be negative; and in 

 general it would appear that the better practical method of apply- 

 ing Sturm's theorem would be not to deal with the Residues, 

 which have hitherto been the sole things considered, but rather 

 with the linear quotients which have been treated as merely inci- 

 dental to the formation of the Residues. 



To find the value of the Rhizoristic series corresponding to a 

 given value of x, the better method would accordingly seem to 

 be to commence with finding the arithmetical values of the (») 

 quotients 



ai.a?+^; agX + b^i . . . ttrtX+bn. 



